Central limit theorems for the real zeros of Weyl polynomials

We establish the central limit theorem for the number of real roots of the Weyl polynomial $P_n(x)=x{i}_0+x{i}_{1}x+\mathrm{.}\mathrm{.}\mathrm{.}+x{i}_n(n!{)}^{(-1\mathrm{/}2)}{x}^n$, where $x{i}_i$ are iid Gaussian random variables. The main ingredients in the proof are new estimates for the correlation functions of the real roots of $P_n$ and a comparison argument exploiting local laws and repulsion properties of these real roots.